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July  2007, 1(3): 371-424. doi: 10.3934/jmd.2007.1.371

Unbounded orbits for outer billiards I

Richard Evan Schwartz 1,

1. 

Department of Mathematics, Brown University, Providence, RI 02912, United States

Received  January 2007 Revised  February 2007 Published  April 2007

The question of B.H. Neumann, which dates back to the 1950s, asks if there exists an outer billiards system with an unbounded orbit. We prove that outer billiards for the Penrose kite, the convex quadrilateral from the Penrose tiling, has an unbounded orbit. We also analyze some finer properties of the orbit structure, and in particular produce an uncountable family of unbounded orbits. Our methods relate outer billiards on the Penrose kite to polygon exchange maps, arithmetic dynamics, and self-similar tilings.
Keywords: arithmetic graph., Penrose kite, unbounded orbits, polygon exchange, outer billiards, dual billiards.
Mathematics Subject Classification: Primary: 37E99; Secondary: 52C2.
Citation: Richard Evan Schwartz. Unbounded orbits for outer billiards I. Journal of Modern Dynamics, 2007, 1 (3) : 371-424. doi: 10.3934/jmd.2007.1.371
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